Algebra Examples

$2 a^{2} b - \frac{5}{2} a b^{2} + \frac{7}{12} a b^{2} + \frac{5}{6} a^{2} b$

Step 1

Simplify each term.

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Step 1.1

Multiply $- \frac{5}{2} (a b^{2})$ .

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Step 1.1.1

Combine $a$ and $\frac{5}{2}$ .

$2 a^{2} b - \frac{a \cdot 5}{2} b^{2} + \frac{7}{12} \cdot (a b^{2}) + \frac{5}{6} \cdot (a^{2} b)$

Step 1.1.2

Combine $b^{2}$ and $\frac{a \cdot 5}{2}$ .

$2 a^{2} b - \frac{b^{2} (a \cdot 5)}{2} + \frac{7}{12} \cdot (a b^{2}) + \frac{5}{6} \cdot (a^{2} b)$

Step 1.2

Move $5$ to the left of $b^{2} a$ .

$2 a^{2} b - \frac{5 \cdot (b^{2} a)}{2} + \frac{7}{12} \cdot (a b^{2}) + \frac{5}{6} \cdot (a^{2} b)$

Step 1.3

Multiply $\frac{7}{12} (a b^{2})$ .

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Step 1.3.1

Combine $a$ and $\frac{7}{12}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{a \cdot 7}{12} b^{2} + \frac{5}{6} \cdot (a^{2} b)$

Step 1.3.2

Combine $\frac{a \cdot 7}{12}$ and $b^{2}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{a \cdot 7 b^{2}}{12} + \frac{5}{6} \cdot (a^{2} b)$

Step 1.4

Move $7$ to the left of $a$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 \cdot a b^{2}}{12} + \frac{5}{6} \cdot (a^{2} b)$

Step 1.5

Multiply $\frac{5}{6} (a^{2} b)$ .

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Step 1.5.1

Combine $a^{2}$ and $\frac{5}{6}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} \cdot 5}{6} b$

Step 1.5.2

Combine $\frac{a^{2} \cdot 5}{6}$ and $b$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{a^{2} \cdot 5 b}{6}$

Step 1.6

Move $5$ to the left of $a^{2}$ .

$2 a^{2} b - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 2

To write $2 a^{2} b$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 a^{2} b \cdot \frac{2}{2} - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 3

Simplify terms.

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Step 3.1

Combine $2 a^{2} b$ and $\frac{2}{2}$ .

$\frac{2 a^{2} b \cdot 2}{2} - \frac{5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{2 a^{2} b \cdot 2 - 5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $a b$ out of $2 a^{2} b \cdot 2 - 5 b^{2} a$ .

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Step 4.1.1

Factor $a b$ out of $2 a^{2} b \cdot 2$ .

$\frac{a b (2 a \cdot 2) - 5 b^{2} a}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 4.1.2

Factor $a b$ out of $- 5 b^{2} a$ .

$\frac{a b (2 a \cdot 2) + a b (- 5 b)}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 4.1.3

Factor $a b$ out of $a b (2 a \cdot 2) + a b (- 5 b)$ .

$\frac{a b (2 a \cdot 2 - 5 b)}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 4.2

Multiply $2$ by $2$ .

$\frac{a b (4 a - 5 b)}{2} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 5

To write $\frac{a b (4 a - 5 b)}{2}$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$\frac{a b (4 a - 5 b)}{2} \cdot \frac{6}{6} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 6

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{a b (4 a - 5 b)}{2}$ by $\frac{6}{6}$ .

$\frac{a b (4 a - 5 b) \cdot 6}{2 \cdot 6} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 6.2

Multiply $2$ by $6$ .

$\frac{a b (4 a - 5 b) \cdot 6}{12} + \frac{7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 7

Combine the numerators over the common denominator.

$\frac{a b (4 a - 5 b) \cdot 6 + 7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $a b$ out of $a b (4 a - 5 b) \cdot 6 + 7 a b^{2}$ .

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Step 8.1.1

Factor $a b$ out of $a b (4 a - 5 b) \cdot 6$ .

$\frac{a b ((4 a - 5 b) \cdot 6) + 7 a b^{2}}{12} + \frac{5 a^{2} b}{6}$

Step 8.1.2

Factor $a b$ out of $7 a b^{2}$ .

$\frac{a b ((4 a - 5 b) \cdot 6) + a b (7 b)}{12} + \frac{5 a^{2} b}{6}$

Step 8.1.3

Factor $a b$ out of $a b ((4 a - 5 b) \cdot 6) + a b (7 b)$ .

$\frac{a b ((4 a - 5 b) \cdot 6 + 7 b)}{12} + \frac{5 a^{2} b}{6}$

Step 8.2

Apply the distributive property.

$\frac{a b (4 a \cdot 6 - 5 b \cdot 6 + 7 b)}{12} + \frac{5 a^{2} b}{6}$

Step 8.3

Multiply $6$ by $4$ .

$\frac{a b (24 a - 5 b \cdot 6 + 7 b)}{12} + \frac{5 a^{2} b}{6}$

Step 8.4

Multiply $6$ by $- 5$ .

$\frac{a b (24 a - 30 b + 7 b)}{12} + \frac{5 a^{2} b}{6}$

Step 8.5

Add $- 30 b$ and $7 b$ .

$\frac{a b (24 a - 23 b)}{12} + \frac{5 a^{2} b}{6}$

Step 9

To write $\frac{5 a^{2} b}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{a b (24 a - 23 b)}{12} + \frac{5 a^{2} b}{6} \cdot \frac{2}{2}$

Step 10

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.1

Multiply $\frac{5 a^{2} b}{6}$ by $\frac{2}{2}$ .

$\frac{a b (24 a - 23 b)}{12} + \frac{5 a^{2} b \cdot 2}{6 \cdot 2}$

Step 10.2

Multiply $6$ by $2$ .

$\frac{a b (24 a - 23 b)}{12} + \frac{5 a^{2} b \cdot 2}{12}$

Step 11

Combine the numerators over the common denominator.

$\frac{a b (24 a - 23 b) + 5 a^{2} b \cdot 2}{12}$

Step 12

Simplify the numerator.

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Step 12.1

Factor $a b$ out of $a b (24 a - 23 b) + 5 a^{2} b \cdot 2$ .

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Step 12.1.1

Factor $a b$ out of $5 a^{2} b \cdot 2$ .

$\frac{a b (24 a - 23 b) + a b (5 a \cdot 2)}{12}$

Step 12.1.2

Factor $a b$ out of $a b (24 a - 23 b) + a b (5 a \cdot 2)$ .

$\frac{a b (24 a - 23 b + 5 a \cdot 2)}{12}$

Step 12.2

Multiply $2$ by $5$ .

$\frac{a b (24 a - 23 b + 10 a)}{12}$

Step 12.3

Add $24 a$ and $10 a$ .

$\frac{a b (34 a - 23 b)}{12}$